On what interval? Are you sure you stated the interval correctly? Was it [a,L) for any a between 0 and L and you restated it (0,L)?

That isn't true on (0,L). It's true on [0,L]. On (0,L) is is a sup, not a max.

I don't know where those last two lines came from. And I would like to see the exact statement of the problem. I don't believe it is true as you have stated it.

You were right in terms of the interval supposed to be closed on [0,L] as well as the exact definition of uniform convergence I just didn't communicate it correctly. If I polished up on my latex it would be easier for me to restate. I attached a jpeg of the question and statement that is confusing me so you can see the exact question.

OK, maybe they meant to write$$
\sup_{(0,L)}\frac N {1+N^2x^2}=N$$The point is that although you have pointwise convergence to ##0## it is not uniform because the larger N is the bigger the sum is near zero. You can't uniformly bound the sum near 0 no matter how large N is.

OK, maybe they meant to write$$
\sup_{(0,L)}\frac N {1+N^2x^2}=N$$The point is that although you have pointwise convergence to ##0## it is not uniform because the larger N is the bigger the sum is near zero. You can't uniformly bound the sum near 0 no matter how large N is.